The influence of injected current on magnetization dynamics is investigated in submicrometer-wide strips of out-of-plane magnetized Pt/Co/GdOx. This ultrathin film structure is similar to Pt/Co/AlOx exhibiting highly efficient current-driven domain wall (DW) motion, which has recently been attributed in part to the Rashba effect. In Pt/Co/GdOx, the reduction of the out-of-plane magnetization with increasing injected current is largely due to Joule heating, and the effective transverse Rashba field that tilts the magnetization is estimated to be at most similar to 0.3 T/(10(12) A/m(2)). Measurements of high-speed DW motion reveal no evidence of a strong Rashba field to suppress DW precession. These findings indicate that the Rashba effect plays a limited or negligible role in current-driven DW dynamics, and alternative torques are required to explain the high efficiency of DW motion in ultrathin heavy-metal/ferromagnet/oxide systems.